During the last reporting period, our research efforts have lead to four publications in scientific journals. These are (in the order given in the bibliography below): 1. A novel way to extract the microscopic mechanism of how proteins acquire their three dimensional structure from kinetic models; 2. A new procedure to analyze nuclear magnetic resonance relaxation experiments to obtain the time scale and amplitude of slow interdomain motions in proteins; 3. A robust method to interpret the colors in the sequence of photons emitted by a single molecule to learn about the molecule's conformational dynamics; 4. Establishing the relation between fluctuations in the intensity of light emitted by dyes attached to a denatured protein and the speed with which this system explores all conformation space. Below, we shall describe the last two projects in more detail. Single-molecule Forster resonance energy transfer (FRET) measurements on freely diffusing molecules contain information about conformational dynamics because the rate of energy transfer depends on the distance between donor and acceptor labels attached to a molecule. In these experiments, a molecule diffuses through a spot illuminated by a laser, and the donor is excited. The output of these experiments is a sequence of photons of different colors (some emitted by the donor and some by the acceptor) separated by apparently random time intervals. As described in previous reports, we have developed a rigorous formal theory that describes how statistics of photons is influenced by protein conformational dynamics, the diffusion of the protein through the laser spot, shot noise, etc. It was shown that the exact FRET efficiency and photon counting histograms can be obtained by solving an appropriate reaction-diffusion equation. We have obtained a simple analytical yet rigorous result for the width of FRET efficiency distribution and showed that the shape of the distribution depends dramatically on the bin size. In reference 3, we introduced and implemented a simpler and more robust procedure to extract the rates of conformational changes by decoding the pattern of colors in the donor/acceptor photon trajectory. For a photon trajectory with measured arrival times, the pattern of colors is analyzed by maximizing the likehood of observing such a trajectory within the framework of a microscopic model. The likelihood function, which must be optimized with respect to the model rate parameters, depends only on how the structure of the molecule changed during the time interval between photons with specified colors. This approach can be applied to bursts of photons emitted by diffusing molecules as well as to long photon trajectories generated by immobilized molecules. This procedure has been illustrated using simulated photon trajectories obtained for systems with two and three different conformational states. Our method works even when the photon colors appear to be random because of high background noise, the photophysical properties of the conformers are similar and/or the photon count rates are similar to the rates of conformational transitions. Because of the clear advantages of this procedure, we expect that it will become widely used. In the above single molecule experiments, because of low detection effiency, the time between observed photons is on the microsecond time scale. Consequently, they contain information only about processes that occur on this and slower time scales. There is a different kind of single molecule FRET experiment, pioneered by Professor B. Schuler of the University of Zurich, in which rare events, where the time interval between photons is on the order of nanoseconds, can be detected. In collaboration with this experimental group, we proposed and implemented (see 4 in the bibliography) a novel and simple procedure to analyze their measured intensity correlation functions. This approach is based on the fact that we were able to find an analytic expression for the conformational correlation time when the dynamics is modelled as diffusion in the presence of the potential of mean force along the interdye distance. We applied this method to experimental donor-donor intensity correlation functions measured for the unfolded subpopulation of a cold shock protein as a function of denaturant concentration. We found that the unfolded protein changes shape on the fifty nanosecond time scale and the more compact the protein, the slower is its internal dynamics. These experiments are new and only very few groups in the world have the expertise to perform them. As technology advances, such experiments will eventually become routine and our procedure will be used to analyze them to learn about a variety of biological processes.